Nanoscale machines are strongly influenced by thermal fluctuations, contrary to their macroscopic counterparts. As a consequence, even the efficiency of such microscopic machines becomes a fluctuating random variable. Using geometric properties and the fluctuation theorem for the total entropy production, a universal theory of efficiency fluctuations at long times, for machines with a finite state space, was developed by Verley et al. [Nat. Commun. 5, 4721 (2014); Phys. Rev. E 90, 052145 (2014)]. We extend this theory to machines with an arbitrary state space. Thereby, we work out more detailed prerequisites for the universal features and explain under which circumstances deviations can occur. We also illustrate our findings with exact results for two nontrivial models of colloidal engines.

We determine the asymptotic forms of work distributions at arbitrary times T, in a class of driven stochastic systems using a theory developed by Nickelsen and Engel (EN theory) [D. Nickelsen and A. Engel, Eur. Phys. J. B 82 , 207 (2011)], which is based on the contraction principle of large deviation theory. In this paper, we extend the theory, previously applied in the context of deterministically driven systems, to a model in which the driving is stochastic. The models we study are described by overdamped Langevin equations and the work distributions in path integral form, are characterised by having quadratic augmented actions. We first illustrate EN theory, for a deterministically driven system - the breathing parabola model, and show that within its framework, the Crooks fluctuation theorem manifests itself as a reflection symmetry property of a certain characteristic polynomial, which also determines the exact moment-generating-function at arbitrary times. We then extend our analysis to a stochastically driven system, studied in references [S. Sabhapandit, EPL 89 , 60003 (2010); A. Pal, S. Sabhapandit, Phys. Rev. E 87 , 022138 (2013); G. Verley, C. Van den Broeck, M. Esposito, New J. Phys. 16 , 095001 (2014)], for both equilibrium and non-equilibrium steady state initial distributions. In both cases we obtain new analytic solutions for the asymptotic forms of (dissipated) work distributions at arbitrary T. For dissipated work in the steady state, we compare the large T asymptotic behaviour of our solution to the functional form obtained in reference [New J. Phys. 16 , 095001 (2014)]. In all cases, special emphasis is placed on the computation of the pre-exponential factor and the results show excellent agreement with numerical simulations. Our solutions are exact in the low noise (beta -> (infinity)) limit.

We obtain exact results for the recently discovered finite-time thermodynamic uncertainty relation, for the dissipated work W-d, in a stochastically driven system with non-Gaussian work statistics, both in the steady state and transient regimes, by obtaining exact expressions for any moment of W-d at arbitrary times. The uncertainty function (the Fano factor of W-d) is bounded from below by 2k(B)T as expected, for all times tau, in both steady state and transient regimes. The lower bound is reached at tau = 0 as well as when certain system parameters vanish (corresponding to an equilibrium state). Surprisingly, we find that the uncertainty function also reaches a constant value at large tau for all the cases we have looked at. For a system starting and remaining in steady state, the uncertainty function increases monotonically, as a function of tau as well as other system parameters, implying that the large t value is also an upper bound. For the same system in the transient regime, however, we find that the uncertainty function can have a local minimum at an accessible time tau(m), for a range of parameter values. The large tau value for the uncertainty function is hence not a bound in this case. The non-monotonicity suggests, rather counter-intuitively, that there might be an optimal time for the working of microscopic machines, as well as an optimal configuration in the phase space of parameter values. Our solutions show that the ratios of higher moments of the dissipated work are also bounded from below by 2k(B)T. For another model, also solvable by our methods, which never reaches a steady state, the uncertainty function, is in some cases, bounded from below by a value less than 2k(B)T.